Time and band limiting operator and Bethe ansatz

TL;DR

This paper explores the spectral properties of the time and band limiting operator using the Bethe ansatz, linking signal processing with quantum integrable systems to provide new insights into their spectra.

Contribution

It introduces a novel application of the Bethe ansatz to diagonalize the Heun operator related to time and band limiting, revealing connections between signal processing and quantum integrable models.

Findings

Spectral connection between time-band limiting and Heun operators

Application of Bethe ansatz to diagonalize the Heun operator

Insights into the spectrum of the time and band limiting operator

Abstract

The time and band limiting operator is introduced to optimize the reconstruction of a signal from only a partial part of its spectrum. In the discrete case, this operator commutes with the so-called algebraic Heun operator which appears in the context of the quantum integrable systems. The construction of both operators and the proof of their commutativity is recalled. A direct connection between their spectra is obtained. Then, the Bethe ansatz, a well-known method to diagonalize integrable quantum Hamiltonians, is used to diagonalize the Heun operator and to obtain insights on the spectrum of the time and band limiting operator.